Worlds of David Darling
Encyclopedia of Science
Home > Encyclopedia of Science

rational number

A number that can be written as an ordinary fraction – a ratio, a/b, of two integers, a and b, where b isn't zero – or as a decimal expansion that either stops (like 4.58) or is periodic (like 1.315315...). Other examples include 1, 1.2, 385.66, and 1/3.

Rational numbers are countable, which means that, although there are infinitely many of them, they can always be put in a definite order, from smallest to largest, and can thus be counted. They also form what's called a densely ordered set; in other words, between any two rationals there always sits another one – in fact infinitely many other ones.

The rational numbers are a subset of the real numbers; real numbers that aren't rational are called, rationally enough, irrational numbers. Although rationals are dense on the real number line, in the sense that any open set contains a rational, they're pretty sparse in comparison with the irrationals. One way to think of this is that the infinity of rationals (which, strangely enough, is exactly the same size as the infinity of whole numbers) is smaller than the infinity of irrational numbers. Another way to grasp the scarcity versus density issue, is to realize that the rationals can be covered by a set whose "length" is arbitrarily small. In other words, given a string of any positive length, no matter how short, it will still be long enough to cover all the rationals. In mathematical parlance, the rationals are a measure zero set. The irrationals, by contrast, are a measure one set. This difference in measure means that the rationals and irrationals are quite different even though a rational can always be found between any two irrationals, and an irrational exists between any two rationals.

Related category